We establish the local $C^{ 1, \alpha}$ regularity of minimizers for functionals of the form $$w\to \int_{ \Omega}(|\nabla w|^p-fw) dx + \int_{ \mathbb{ R}^n}\int_{ \mathbb{ R}^n} \frac{ |w(x)-w(y)|^q}{ |x-y|^{ n+sq}}dx\, dy,$$ where $s \in (0, 1)$, $1 < p \leq sq$, and $f \in L^\infty(\Omega)$. This result complements the work of De Filippis and Minigione in \cite{ DFM}, thereby completing the proof of $C^{ 1,\alpha}$ regularity for all $p, q \in (1, \infty)$ and $s \in (0, 1)$ with locally bounded source term.
Categories: math.AP